What's the connection between even/odd harmonics and even/odd (transfer) functions?
Why do they correspond to each other?
Particularly, if the transfer function is/contains even or odd functions, then why does this lead to the output signal having even or odd functions correspondingly?
Harmonics are the components of a signal that exist as integer multiples of the signal. Even harmonics are: $2f, 4f, ...$, odd are: $1f, 3f, ...$.
A transfer function is some sort of "mapping" that usually applies some sort of "effect" to a signal, e.g. an audio signal. It's often a Fourier transform of some input function.
There are only so many words available to scientists and mathematicians. Sometimes the same words are appropriated for many different purposes. In this case, "even--odd" is appropriated for many different dichotomies: even and odd integers; even and odd functions; even and odd harmonics. From a linguistic point of view, it might be fun to think about whether there are any more mathematical uses of "even--odd" to represent a dichotomy. But from a mathematical point of view, there's no real significance.